Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 107 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 33 + 83\cdot 107 + 52\cdot 107^{2} + 57\cdot 107^{3} + 61\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 + 13\cdot 107 + 101\cdot 107^{2} + 106\cdot 107^{3} + 74\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 56 + 102\cdot 107 + 25\cdot 107^{2} + 99\cdot 107^{3} + 68\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 + 58\cdot 107 + 25\cdot 107^{2} + 65\cdot 107^{3} + 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 69 + 39\cdot 107 + 61\cdot 107^{2} + 49\cdot 107^{3} + 68\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 93 + 14\cdot 107 + 34\cdot 107^{2} + 57\cdot 107^{3} + 8\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 97 + 42\cdot 107 + 50\cdot 107^{2} + 93\cdot 107^{3} + 2\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 100 + 72\cdot 107 + 76\cdot 107^{2} + 5\cdot 107^{3} + 34\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,7)(3,8)(5,6)$ |
| $(1,2,6,3)(4,8,5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,3)(4,5)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,7)(3,8)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,5)(3,4)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,2,6,3)(4,8,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
